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Sparse representations of signals, in particular with learned dictionaries, are widely used for state-of-the-art audio, image, and video restoration. In this paper, the problem of denoising and occlusion restoration of 3D range data based on dictionary learning and sparse representations is explored. We consider the 3D surface obtained from a desktop range scanner as an image, where the value of each pixel represents the depth of a point on the 3D surface. Having this image, we apply techniques from dictionary learning and sparse representation to enhance the acquired 3D surface. These techniques use the spare decomposition of the overlapping patches in the image, over an adapted over-complete dictionary, for enhancing the data. We present experimental results of denoising 3D surfaces following this approach. We also propose an algorithm for filling the missing information regions on 3D scans and demonstrate its effectiveness. Our experimental results are on range data obtained from a low-cost structured-light range scanner.

A geometric framework for the recognition of three-dimensional objects represented by point clouds is introduced in this paper. The proposed approach is based on comparing distributions of intrinsic measurements on the point cloud. In particular, intrinsic distances are exploited as signatures for representing the point clouds. The first signature we introduce is the histogram of pairwise diffusion distances between all points on the shape surface. These distances represent the probability of traveling from one point to another in a fixed number of random steps, the average intrinsic distances of all possible paths of a given number of steps between the two points. This signature is augmented by the histogram of the actual pairwise geodesic distances in the point cloud, the distribution of the ratio between these two distances, as well as the distribution of the number of times each point lies on the shortest paths between other points. These signatures are not only geometric but also invariant to bends. We further augment these signatures by the distribution of a curvature function and the distribution of a curvature weighted distance. These histograms are compared using the @g^2 or other common distance metrics for distributions. The presentation of the framework is accompanied by theoretical and geometric justification and state-of-the-art experimental results with the standard Princeton 3D shape benchmark, ISDB, and nonrigid 3D datasets. We also present a detailed analysis of the particular relevance of each one of the different proposed histogram-based signatures. Finally, we briefly discuss a more local approach where the histograms are computed for a number of overlapping patches from the object rather than the whole shape, thereby opening the door to partial shape comparisons.

In this note we propose to introduce physical constraints in the localization problem in sensor networks. This is based on extending the classical STRESS function from distance geometry and multidimensional scaling. We present the underlying framework and demonstrate its importance with three examples: penalizing the sensors for being in high elevation areas, removing sensors from forbidden areas, and forcing the sensors to be on pre-described curves. We also extend the work to dynamic environments.

In this note we show how to introduce physical constraints in the localization problem in sensor networks. This is based on extending the classical STRESS function from distance geometry and multidimensional scaling. We present the underlying framework and demonstrate it with three examples: penalizing the sensors for being in high elevation areas, removing sensors from forbidden areas, and forcing the sensors to be on pre-described curves.